Naturally, this rule is meaningless to those who espouse the frequency interpretation of probability, for whom probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon a state of information.

- "
*p*+_{2}*p*> 0.9"_{3}

Given testable information, the maximum entropy procedure consists of seeking the probability distribution which maximizes information entropy, subject to the constraints of the information. This constrained optimization problem is typically solved using the method of Lagrange multipliers.

Entropy maximization with no testable information takes place under a single constraint: the sum of the probabilities must be one. Under this constraint, the maximum entropy probability distribution is the uniform distribution,

The λ_{k} parameters are Lagrange multipliers whose particular values are determined by the constraints according to

For continuous distributions, (Jaynes, 1963, 1968, 2003) finds that the limiting form of the entropy expression as the distribution approaches a continuous distribution is

We have some testable information *I* about a quantity *x* which takes values in some interval of the real numbers (all integrals below are over this interval). We express this information as *m* constraints on the expectations of the functions *f _{k}*, i.e. we require our epistemic probability density function to satisfy

The following argument is the result of a suggestion made by Graham Wallis to E. T. Jaynes in 1962 (Jaynes, 2003). It is essentially the same mathematical argument used for the derivation of the partition function in statistical mechanics, although the conceptual emphasis is quite different. It has the advantage of being strictly combinatorial in nature, making no reference to information entropy as a measure of 'uncertainty', 'uninformativeness', or any other imprecisely defined concept. The information entropy function is not assumed *a priori*, but rather is found in the course of the argument; and the argument leads naturally to the procedure of maximizing the information entropy, rather than treating it in some other way.

Suppose an individual wishes to make an epistemic probability assignment among *m* mutually exclusive propositions. She has some testable information, but is not sure how to go about including this information in her probability assessment. She therefore conceives of the following random experiment. She will distribute *N* quanta of epistemic probability (each worth 1/*N*) at random among the *m* possibilities. (One might imagine that she will throw *N* balls into *m* buckets while blindfolded. In order to be as fair as possible, each throw is to be independent of any other, and every bucket is to be the same size.) Once the experiment is done, she will check if the probability assignment thus obtained is consistent with her information. If not, she will reject it and try again. Otherwise, her assessment will be

Now, in order to reduce the 'graininess' of the epistemic probability assignment, it will be necessary to use quite a large number of quanta of epistemic probability. Rather than actually carry out, and possibly have to repeat, the rather long random experiment, our protagonist decides to simply calculate and use the most probable result. The probability of any particular result is the multinomial distribution,

The most probable result is the one which maximizes the multiplicity *W*. Rather than maximizing *W* directly, our protagonist could equivalently maximize any monotonic increasing function of *W*. She decides to maximize

Jaynes, E. T., 1963, `Information Theory and Statistical Mechanics', in Statistical Physics, K. Ford (ed.), Benjamin, New York, p. 181. Available here.

Jaynes, E. T., 1968, `Prior Probabilities', IEEE Trans. on Systems Science and Cybernetics, SSC-4, 227. Available here.

Jaynes, E. T., 2003, 'Probability Theory: The Logic of Science', Cambridge University Press, 2003.